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wherefore (IV. 3.) OG is equal to the greater segment of side AO divided by a medial section. But (II. 20.) the square of AC, drawn from the vertex to a point in the extension of the base of the triangle OAG, is equivalent to the square of AG, together with the rectangle under OC and CG, or the square of OG ; that is, the square of the side of the inscribed pentagon is equivalent to the squares of AO and of AB, the sides of the hexagon and decagon. Cor. The triple chord AD of the decagon is equal to the combined sides AO and AB of the inscribed hexagon and decagon. For the triangle OAG, being equal to AOB or COD, the angle DCO or DCG is equal to AGO or DGC, and consequently (I. 11.) CD is equal to GD. Wherefore AD being equal to AG and GD, is equal to AO with OG or AB. Scholium. Hence the sides of the inscribed decagon and pentagon may be found by a single construction. For draw the perpendicular diameters AC and EF, bisect OC in D, join DE, make DG equal to it, and join GE. It is evident, that AO is cut medially in G (II. 19.), and consequently that OG is equal to a side of the inscribed decagon. But GOE being a right- angled triangle, the square of GE is equivalent to the squares of GO and OE (II. 10.), or the squares of the sides of the decagon and hexagon; whence GE is equal to the side of the inscribed pentagon. It also follows, that CG is equal to CI or CP, the triple chords of the inscri

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PROP. XX. PROB.

In a given circle, to inscribe regular polygons of fifteen and of thirty sides.

Let AB and BC be the sides of an inseribed decagon, and AD the side of a hexagon inscribed; the arc BD will be the fifteenth part of the circumference of the circle, and DC the thirtieth part.

For, if the circumference were divided into thirty equal portions, the arc AB would be equal to three of these, and the : D. arc AD to five; consequently the B excess BD is equal to two of these portions, or it is the fifteenth part & of the whole circumference. Again, the double arc ABC being equal to six portions, and ABD to five, the defect DC is equal to one portion, or to the thirtieth part of the circumference.

Scholium. From the inscription of the square, the pentagon, and the hexagon,-may be derived that of a variety of other regular polygons: For, by continually bisecting the intercepted arcs and inserting new chords, the inscribed figure will, at each successive operation, have the number of its sides doubled. Hence polygons will arise of 6, 8, and 10 sides; then of 12, 16, and 20 ; next of 24, 32, and 40; again, of 48, 64, and 80; and so forth repeated. ly. The excess of the arc of the hexagon above that of the decagon, gives the arc of a fifteen-sided figure; and the continued bisection of this arc will mark out polygons with 30, 60, or 120 equal sides, in perpetual succession.

The same results might also be obtained from the differences of the preceding arcs. Of the regular polygons, three only are susceptible of perfect adaptation, and capable therefore of covering, by their repeated addition, a plane surface. These are the equilateral triangle, the square, and the hexagon. The angles of an equilateral triangle are each two-thirds of a right angle, those of a square are right angles, and the angles of a hexagon are each equal to four-third parts of a right angle. Hence there may be constituted about a point, six equilateral triangles, four squares, and three hexagons. But no other regular polygon can admit of a like disposition. The pentagon, for instance, having each of its angles equal to six-fifths of a right angle, would not fill up the whole space about a point, on being repeated three times; yet it would do more than cover that space, if added four times. On the other hand, since each angle of a polygon which has more than six sides must exceed four-third parts of a right angle, three such polygons cannot stand round a point. Nor can the space about a point ever be bisected by the application of any regular polygons, of whatever number of sides; for their angles are always necessarily each less than two right angles.

ELEMENTS

Os

G E O M ETR Y.

BOOK W.

OF PROPORTION,

THE preceding Books treat of magnitude as concrete, or having mere extension; and the simpler properties of lines, of angles, and of surfaces, were deduced, by a continuous process of reasoning, grounded on the principle of superposition. But this mode of investigation, how satisfactory soever to the mind, is by its nature very limited and laborious. By introducing the idea of Number into geometry, a new scene is opened, and a far wider prospect rises into view. Magnitude, being considered as discrete, or composed of integrant parts, becomes assimilated to multitude ; and under this aspect, it presents a vast system of rela

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